So I screwed up a problem on my exam. I know that now. But pure mathematics is as difficult and terrifying as it is rewarding for me, and I can't let this go! If someone could tell me if the following is right or wrong, that would be much appreciated.

I was asked to prove the uniform convergence of the series $f_n(x)=\frac{x}{x+n}$ on $[1,\infty)$.

Not really, sorry. You need to show, for given $\varepsilon>0$, that there is $N\in \mathbb{N}$, such that $$\frac{x}{x+n}< \varepsilon$$
for all $n\ge N$, independently of $x$. Which, I'm afraid, is not even true, because if you have any $N$ you can choose $x $ very large (larger than $N$) such that the fraction becomes $>\frac{1}{2}$, say. What can be shown is uniform convergence on compact sets, then you'd start out with an upper bound on $x$